A generic condition implying o-minimality for restricted C<span class="mathjax-formula">$^{\infty }$</span>-functions

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

OLIVIERLEGAL

A generic condition implying o-minimality for restricted C^∞-functions

Tome XIX, n^o3-4 (2010), p. 479-492.

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A generic condition implying o-minimality for restricted C

-functions

Olivier Le Gal⁽¹⁾

ABSTRACT.— We prove that the expansion of the real field by a restricted C^∞-function is generically o-minimal. Such a result was announced by A. Grigoriev, and proved in a different way. Here, we deduce quasi-analyti- city from a transcendence condition on Taylor expansions. This then im- plies o-minimality. The transcendance condition is shown to be generic.

As a corollary, we recover in a simple way that there exist o-minimal struc- tures that doesn’t admit analytic cell decomposition, and that there exist incompatible o-minimal structures. We even obtain o-minimal structures that are not compatible with restricted analytic functions.

R^´ESUM´E.— On montre que g´en´eriquement, l’expansion du corps des r´eels par une fonction C^∞restreinte est o-minimale. Un r´esultat du mˆeme type utilisant d’autres d’arguments a ´et´e annonc´e par A. Grigoriev. Ici, nous utilisons une condition de transcendance sur les d´eveloppements de Taylor pour assurer la quasianalyticit´e de certaines alg`ebres diff´e- rentielles, ce qui implique la o-minimalit´e. On montre que cette condi- tion de transcendance est g´en´erique. Comme corollaire de ce r´esultat, on donne des preuves simples du fait qu’il existe des structures o-minimales n’admettant pas de d´ecomposition cellulaire analytique, et qu’il existe des structures o-minimales incompatibles. On obtient mˆeme des structures o-minimales non compatibles avec les fonctions analytiques restreintes.

(∗) Re¸cu le 11/03/2009, accept´e le 27/04/2009

(1) Dpto. Algebra, Geometr´ıa y Topologia. Facultad de Ciencas. E-47005-Valladolid.

Spain.

[email protected]

1. Introduction

The geometry of the zero set of a C^∞-function may be very complicated, since Whitney proves that any closed subset ofRⁿ is such a zero set.Ren´e Thom deep interest on genericity was driven by the expectation that the objects coming from generic smooth mappings should have atamegeometry.

On the other hand, the axiomatic approch of o-minimality garantee such a tame geometric behavior.In [Gri05], A.Grigoriev announces that in one sense, these two approaches meet together : o-minimality is in a certain meaning generic.This article aims to give another version of this result, based on the notion of quasianalyticity.

Recall that an expansionRF = (R,0,1,+,·, <,F) of the real field by a family of maps F is o-minimal if any definable set has finitely many connected component.Geometrically speaking, a setX ⊂R^k isdefinable in R_F if it belongs to the smallest collection of subsets of Rⁿ, for various n, which contains semi-algebraic sets and all the graphs of maps inF, and is closed under basic set theoretic operations : finite boolean combinations, cartesian products and projections.A map is also said to be definableif its graph is definable.

O-minimality provides many geometric properties for definable sets or maps, such as Whitney stratifications.Some structures verify an addition- nal tameness property, that imply, for instance, 8Lojaciewicz inequalities:RF

is polynomially boundedif for any definable functionf :R→R, there exists an integernsuch thatf(x)/xⁿis ultimately bounded.For a more pre- cise introduction to o-minimal geometry, we refer to the books [Cos00] of Coste and [vdD98] of van den Dries.Among others, the following are clas- sical examples of o-minimal structures.Using the Gabrielov’s theorem of the complement [Gab96], Denef and van den Dries prove that the structure Rangenerated by the familyanof restricted analytic functions is o-minimal and polynomially bounded [DvdD88].The same holds for certain structures generated by quasi-analytic Denjoy-Carleman classes, according to the re- sult [RSW03] of Rolin, Speissegger and Wilkie.Wilkie also prove that the structureRPfaff, generated by the so-called pfaffian functions, is o-minimal [Wil99].

We fix some notations.Lethbe aC^∞-function in an open subsetU ofR. Themulti-jetj_n^mh(x) of ordermofhat then-tuplex= (x₁, . . . , x_n)∈ Uⁿ is the n(m+ 1)-tuple of the values of h and its m first derivatives at the pointsx1, . . . , xn:j^m_nh(x) = (h(x1), . . . , h(xn), . . . , h^(m)(x1), . . . , h^(m)(xn)).

We definehto bestrongly transcendentalif for any finite subset{x1, . . . , xn} of U, the values of h and all its derivatives at x1, . . . , xn verify with x1, . . . , xn at most a finite number of integral algebraic relations.Precisely,

if ∆_n ={(x₁, . . . , x_n)∈Rⁿ; ∃i=j, x_i =x_j} denotes the diagonals, his strongly transcendental if

∀n1, ∀x= (x1, . . . , xn)∈ Uⁿ\∆n, ∃C∈N, ∀m∈N,

trdeg(x, j^m_nh(x))n(m+ 2)−C, (1.1) where trdeg denotes the transcendence degree overQ.We say that a function h: [0,1]→Ris arestricted C^∞-function – denoted byh∈C^∞([0,1]) – if it is the restriction of some C^∞-functionhdefined in a neighborhood of [0,1].If h can be chosen to be strongly transcendental, we say thathis a restricted strongly transcendental function.

We now set our main results.Recall that the weak Whitney topology – induced by the family of seminorms sup_t∈K|h^(k)(t)|, whereKranges in the compact subsets of U – turns C^∞(U) into a Baire space: anyresidualset – meaning that it contains a countable intersection of open dense sets – is dense.

Theorem 1.1. — The set stof restricted strongly transcendental func- tions is residual in C^∞([0,1]).

The proof is presented in section 2.The interest of restricted strongly transcendantal functions appears in the following.

Theorem 1.2. — Lethbe a restricted strongly transcendental function.

Then, Rh is o-minimal, polynomially bounded and admits C^∞-cell decom- position.

The scheme of the proof of this theorem is merely an adaptation of the methods developed in [LGR08] to construct an o-minimal structure that does not admit C^∞-cell decomposition, joint with the main theorem of [RSW03], which asserts that o-minimality follows from quasi-analyticity.

We have devoted section 3 to this proof.

In the last section, we prove as corollaries that there exist o-minimal structures that does not admit analytic cell decomposition, and that there exists a pair of functions (h1, h2) such thatRh1 andRh2 are o-minimal but Rh1,h2 is not.Such results where first obtained in [RSW03], with the use of quasi-analytic Denjoy-Carleman classes.Here, we can improve the last, by imposing one of the two functions to be analytic.It gives rise to the following, which seems to be actually unknown :

Corollary 1.3. — There exists an o-minimal expansionRh of the real field such that Rh,an is not o-minimal.

We conclude with some questions that arise from this work.While they are generic, we dont explicitly know any strongly transcendental function.

Is it possible to give the construction of such a function? On another hand, one could easily enhance our result for expansions of (R,+,·, <) by finitely many functions, meaning that the k-tuples (h₁, . . . , h_k) of restricted C^∞- functions such that Rh1,...,hk is o-minimal are generic among k-tuples of restricted C^∞-functions.Does such a result hold for infinite families? The question becomes complicated even to formulate, since it is not clear what topology should be considered on the set of all families of restricted C^∞- functions.

Notations.— Throughout this paper,Ndenotes the set of natural num- bers andRthe real field.The lettersi, j, k, , m, n, pandqdenote non nega- tive integers,xthen-tuple (x₁, . . . , x_n), andta single real variable.The let- tersf andgdenote maps or their germs, andha one variableC^∞-function.

We use the notationstfor the set of restricted strongly transcendental func- tions, andanfor restricted analytic functions.

2. Genericity and Transcendence

In this section, we show the genericity of the transcendance condition (1).Recall that the transcendence degree of a k-tuple (x₁, . . . , x_k)∈R^k is the minimum of the dimension of any integral algebraic subsets ofR^k that contains the point (x₁, . . . , x_k).Lethbe aC^∞-function on a open bounded subsetU ofR.Saying thathis strongly transcendental then means that for any givenx∈ Uⁿ\∆n, the codimension of the integral algebraic subsets of R^n(m+2)that contain the point (x, j_n^mh(x)) is ultimately bounded whenm tends to infinity.Actually, we will show a stronger result: the bound on this codimension is genericallyn.The bound is in particular uniform inx.

Lemma 2.1. — Let U be an open bounded subset in R, and X be an algebraic set of codimension n+ 1 inR^n(m+2). Then, the set E(X) ={h∈ C^∞(U);∀x∈ Uⁿ\∆_n, (x, j_n^mh(x))∈/X} is residual.

Proof. —Fix an algebraic set X ⊂ R^n(m+2) of codimension n+ 1. To prove the lemma, we introduce the followings:

Ui ={x1∈ U; d(x1,Fr(U) 1

i}, ∆_n,i={x∈ Uⁿ;d(x,∆_n)<1 i}, Ei(X) ={h∈C^∞(U);∀x∈ U_iⁿ\∆n,i, (x, j_n^mh(x))∈/X}, whereddenote the euclidian distance and Fr(U) the frontierU \ U ofU.

Let us show that E_i(X) is a dense open subset of C^∞(U).It is open because of the compactness ofUiⁿ\∆_n,i.Indeed, if (h_i)_i∈N is a convergent sequence in the complement ofE_i(X) with limith, we can define a sequence (x_i)_i∈NinUiⁿ\∆_n,i such that (x_i, j_n^mh_i(x_i))∈X.The sequence (x_i) accu- mulates to a pointxinU_iⁿ\∆n,i.SinceXis closed, we have (x, j_n^mh(x))∈X, so that his in the complement of E_i(X).So, the complement of E_i(X) is closed, andEi(X) is open.

In order to prove thatEi(X) is dense, we fix a functionh∈C^∞(U), and we construct a small perturbation ofhthat belongs toEi(X).We obtain this perturbation by adding a polynomial to h.For ε= (ε1, ε2, . . . , εn(m+1))∈ R^n(m+1), we set

hε(t) =h(t) +ε1t^n(m+1)−1+ε2t^n(m+1)−2+. . .+εn(m+1)−1t+εn(m+1)

and we consider the map

ϕ: U_iⁿ\∆_i,n × R^n(m+1) → U_iⁿ×R^n(m+1)

x ε → x, j_n^mhε(x)

With this notation, the function h_ε belongs to E_i(X) if for all x ∈ U_iⁿ \

∆i,n, the point ϕ(x, ε) does not belong to X.Hence, we require an ε as small as desired such that ε lie in the complement of π(ϕ⁻¹(X)), where π:R^n(m+2)→R^n(m+1) denotes the projectionπ(x, ε) =ε.

SinceXis an algebraic set of codimensionn+1, it is the union of finitely many smooth manifolds of codimension at least n+ 1.Moreover, the map ϕis a diffeomorphism; it is bijective and has constant rankn(m+ 2).The pre-image ϕ⁻¹(X) is then the union of finitely many smooth manifolds of codimension at leastn+ 1.The projection of each of these manifolds byπ might be singular, but according to the Sard theorem, it has measure zero.

Hence π(ϕ⁻¹(X)) is the union of finitely many sets of measure zero, then has measure zero itself.

Let us return to the perturbationhε.We have shown that the set of ε such that hε does not belong to Ei(X) has measure zero.This set cannot contain any neighborhood of 0.Hence, there exists ε as small as desired such thathεbelongs toEi(X), which shows thatEi(X) is dense.

Now, remark thatE(X) =

i∈N^∗E_i(X).Since for alli∈N^∗, E_i(X) is an open dense set,E(X) is residual, which achieves the proof.

We can now prove theorem 1.1.

Proof. —[Proof of Theorem 1.1] We first show that strongly transcen- dantal functions are residual in C^∞(U), where U denote a bounded open

subset ofR.LetE be the set

E={h∈C^∞(U); ∀(n, m)∈N²,for all integral algebraicX ⊂R^n(m+2), Codim(X)n+ 1⇒ ∀x∈ Uⁿ\∆n, (x, j^m_nh(x))∈/ X}.

First remark that any function inEis strongly transcendental.Indeed, if hbelongs toE, the transcendence degree of (x, j_mⁿh(x)) is at leastn(m+ 1) for any pointxinUⁿ\∆_n.The transcendence condition (1) is then satisfied with C =n.Moreover, the set E is the intersection of all E(X), when X ranges over all integral algebraic subsets ofR^n(m+2)of codimension at least n+ 1, for all integersn, m.There are countable many such (n, m, X), and according to lemma 2.1, anyE(X) is residual.ThenEis residual also.Since it containsE, the set of strongly transcendental functions inU is residual.

Now, fix a bounded neighborhood U of [0,1], and denote by r the re- striction mapr:h∈C^∞(U)→h|[0,1]∈C^∞([0,1]).The mapris surjective (any restricted C^∞ function can be extended toU), linear and continuous.

So, by the Open Mapping Theorem, r is an open map, then r concerves genericity.Sincer(E)⊂st, restricted strongly transcendental functions are generic in restricted C^∞-functions as well.

Actually, the same ideas show a slightly different result.Let R be a positive real number, and denote byAR the subset of analytic functions in U given by:

AR={f ∈C^∞(U); ∃C∈R, ∀k0,∀x∈ U,|f^(k)(x)|CR^kk!}.

The R-norm ||f||R = sup_k∈N,x∈U|^f_R^(k)k^(x)k! | turns AR into a Banach space.

The following proposition claims that strongly transcendantal functions are also generic in any affine subspace of C^∞(U) of directionAR.It will be used in the proof of corollary 1.3.

Proposition 2.2. — Let U be an open bounded set, h ∈ C^∞(U), and R >0. Then strongly transcendental functions are generic inh+AR, with respect to the topology induced onh+AR by theR-norm onAR.

Proof. — We associate to any algebraic setX⊂R^n(m+2)of codimention at last n+ 1 the set E_h(X) = E(X)∩(AR+h) = {g ∈ (AR+h);∀x∈ U \∆_n,(x, j_n^mg(x))∈/ X}.Again, E_h(X) is residual inAR+h: since the perturbation used in the proof of lemma 2.1 was obtained by adding a polynomial – which belongs toAR and can be choosen to have anR-norm as small as desired – the same proof holds.Now, since the set of strongly transcendantal functions in AR+hcontains the intersection of allEh(X)

whenX ranges over all integral algebraic subsets ofR^n(m+2)of codimention n+ 1, for alln, m, strongly transcendantal functions are generic inAR+h.

3. Transcendence and o-minimality

This section is devoted to the proof of theorem 1.2. The principle is close to a reasoning made in [LGR08] (see part A.). The o-minimality is obtained as a consequence of the main theorem of [RSW03] : RF is o-minimal pro- vided that the algebras generated by F are quasianalytic.Here generated means with respect to compositions, implicit functions, monomial divisions as well as algebraic operations.In a first step, we introduce a germified ver- sion of those algebras, and a family of operators that act naturally on it – namely, the operators corresponding to take implicit functions, to divide by a monomial, to compose and to make algebraic operations.In a second step, we recall two elementary lemmas, which make the action of these operators on Taylor series more precise.Finally, we use those properties to deduce the quasianalyticity of the algebras from strong transcendence.

3.1. Operators, algebras

Our goal is to introduce theorem 3.3, which is a slightly modified version of theorem 5.2 of [RSW03]. It gives a sufficient condition for o-minimality.

We first define similar algebras than in [RSW03], together with similar op- erators than in [LGR08].Denote by Fn the algebra of the germs at 0 of maps in C^∞(Rⁿ).

Definition 3.1. — We call elementary operators the following op- erators:

1. Constant operators (of arity0), defined for each polynomial P∈R[x]

by:

→P(x)∈ Fn ;

2. Compositions, defined for (n, m) ∈N² on Fn× {(g1, . . . , gn)∈ Fmⁿ; gi(0) = 0, i= 1, . . . , n}by :

f, g1, . . . , gn→f(g1, . . . , gn)∈ Fm;

3. Monomial divisions, defined forn∈N^∗on{f ∈ Fn; f(x₁, . . . , x_n−1,0)

= 0} by:

f →g∈ Fn, withg= f

x_n ifxn= 0, andg=∂nf otherwise;

4. Implicit function operators, defined forn∈N^∗ on{f ∈ Fn+1; f(0) = 0, ∂_n+1f(0) = 1} by:

f →ϕ∈ Fn, withf(x₁, . . . , x_n, ϕ(x₁, . . . , x_n)) = 0.

These operators can be composed ones to another as soon as the image of the ones are in the definition set of the other. We call operatorsuch a composition.

We callBorel map– denoted by– the application that maps a C^∞- germ at 0 to its Taylor expansion.Remark that for any operatorL, the Tay- lor series ofL(f) only depends on the Taylor series off.We can then extend the Borel map to operators: any operatorL admits aformal conterpart Lthat verifiesL(f₁, . . . ,f) =(L(f₁, . . . , f)).For a givenh∈C^∞(U), we now define the algebras generated by h, to be the collection of the germs obtained by the action of the operators on the germ ofhat some points of U.Precisely :

Definition 3.2. — Leth be aC^∞-function on an open setU. The al- gebra generated by h is the collection A(h) = (An(h); n ∈ N) of the smallest algebrasAn(h)⊂ Fn such that:

1. For alla∈ U, the germ at 0of t→h(a+t)belongs toA1(h);

2. If L : Fn1×. . .× Fni → Fm is an operator, and if (f1, . . . , fi) ∈ An1(h)×. . .× Ani(h) belongs to the definiton set of L, then L(f1, . . . , fi) belongs toAm(h).

Let us introduce a version of theorem 5.2 of [RSW03] convenient to our propose.Recall that an algebraG ⊂ Fn is said to bequasi-analyticif the Borel map restricted toG is injective.

Theorem 3.3 (Rolin, Speissegger, Wilkie.See 5.2 in [RSW03]).—Let U ⊂Rbe open andh∈C^∞(U). Denote byHthe collection of all the restric- tions ofhto a closed interval inU. If the algebrasAn(h)are quasi-analytic, R_His o-minimal, polynomially bounded, and admits C^∞-cell decomposition.

Proof. — Denote by C = {CB; B compact box ofRⁿ, n ∈N} the col- lection of the subalgebrasCB ofC^∞(B) obtained by the following way.

1.For all compact box B⊂ U, h|B belongs toCB;

2.For every compact boxB ⊂Rⁿ,CB containsR[x₁, . . . , x_n];

3.Ifg= (g₁, . . . , g_n)∈ C_Bⁿ,f ∈ CB andg(B)⊂B, thenf◦g∈ CB; 4.IfB⊂B andf ∈ CB,f|B∈ CB;

5. ∂f /∂xi∈ CB for everyf ∈ CB;

6.Ifn >1 andf ∈ CBis such thatf(0) = 0 and (∂f /∂xn)(x)= 0 for all x∈ CB, then the mapαgiven onBbyf(x1, . . . , xn−1, α(x1, . . . , xn−1))

= 0 belongs toCB, whereBis the canonical projection ofBonRⁿ⁻¹; 7.Iff ∈ CB andf /xi admits aC^∞ extensiong onB, then g∈ CB. Remark that C verifies the conditions (C1)–(C4)(C6)(C7) of [RSW03]

(the extension part of condition (C3) holds because any compact boxB ⊂ U can be extended to a compact box B with B ⊂ Int(B) ⊂ B ⊂ U).

Theorem 3.3 then holds provided that the condition C5 of [RSW03] holds, this means that the quasianalyticity ofAn(h) implies the quasianalyticity of the algebras Cn of [RSW03] (Cn is the algebra of the germs at 0 of the maps in CB for all box B with 0 ∈ Int(B)).It then suffices to check that the germ at 0∈Rⁿ of any functionx→f(a+x) belongs toAn(h) for any f ∈ CB and anya∈B⊂Rⁿ.

Each functionf in CB is obtained from (1) and (2) by applying finitely many operations from (3)–(7).We proceed by induction on the number of operations needed to construct f.If no operation is needed, the germ of x → f(a+x) is in A(h) either by applying a constant operator, or because of the first condition in the definition of A(h).If f is obtained by (3) the claim holds by applying the composition operator.If f = ∂g/∂xi

is obtained by (5), the germ at 0 ∈ Rⁿ of x → f(a+x) is obtained as [Dxn+1(ga(x1, . . . , x_i−1, xi+xn+1, xi+1, . . . , xn)−ga]◦φwheregais the germ of x→g(a+x),g_a is the germ (x, x_n+1)→g_a(x),D_xn is the operator of division byx_n+1andφ₁is the germ ofx→(x₁, . . . , x_n,0).Iffis obtained by (6), remark thatA(h) is closed by taking implicit function with last partial derivative not necessary 1, by multiplying by a constant before applying the operator.Finally, the germ of a monomial divisiong(x)/x_i is obtain at x_i= 0 by the corresponding operator, and atx_i =a_i = 0 by applying the implicit function operator to (x, xn+1)→g(a+x)−xn+1(ai+xi).

3.2. Two Lemmas on operators

According to theorem 3.3, a restricted strongly transcendantal function his o-minimal as soon as the algebrasAn(h) are quasianalytic.In order to

obtain this quasi-analyticity, we need two properties, that were introduced in [LGR08].The following express the algebraic behavior of the formal op- erators.We denote by |β| = β₁+. . .+β_k the norm of the muti-index β= (β₁, . . . , β_k).

Lemma 3.4. — Let L: Fi1, . . . ,Fim → Fn be an operator. Then, there exists (a₁, . . . , a_k) ∈ R^k, and for all α ∈ Nⁿ, ∈ {1, . . . , m}, there ex- ist β_,α ∈ Nⁱ and P_α ∈ Q[x₁, . . . , x_{k+|β1,α|+...+|βm,α|}] such that, for all (f₁, . . . , f_m)in the definition domain ofL, we have

L(f1, . . . ,fm) =

α∈Nⁿ

Pα(a1, . . . , ak, f1,0, . . . , f1,β1,α, . . . , fm,0, . . . , fm,βm,α)x^α

wherefp=

β∈N^ipfp,βx^β.

Proof. — One can check that the statement of the lemma is preserved by composition: ifL,M1, . . . ,Mjverify the conclusion of 3.4 then also does L(M1, . . . ,Mj) (see [LGR08] for more details).Hence, by an induction on the number of operators which appear in the definition ofL, the lemma is true if it holds for elementary operators.Now, the statement is obvious for constant operators and monomial divisions (coefficients a1, . . . , ak appear for constants operators.They are the coefficients of the polynomial).An easy computation shows that it holds also for composition operators.At last, if L is an implicit function operator, the coefficients of L(f) are obtained from those of fby solving a triangular system, with coefficients 1 on the diagonal : recall that we defineLonly for mapsf with∂n+1f(0) = 1.Hence, the maps Pα of the statement are actually polynomials, and not rational

maps.

The following is a quasi-analyticity property for operators.

Lemma 3.5. — LetL be an operator. Then, L= 0 implyL= 0.

Proof. — This is lemma 2.4 in [LGR08]. We recall the main steps.

First remark that the operators are continuous.Precisely, letL:Fi1× . . .×Fim → Fnbe an operator,F = (f₁, . . . , f_m) be maps whose germs at 0 belong to the definition set ofL.Then there exist a compact neighborhood U of 0∈R^i1+...+im, a compact neighborhoodV of 0∈Rⁿ, a neighborhood W ofF in C^∞(U,R^k) and a well defined operatorL onW, continuous for theC^∞-topology on C^∞(U) and C^∞(V), which acts on the elements ofW as L does on germs.This is obtained by an induction on the number of

elementary operators which are involved in L.Continuity remains true by induction, and is a well known fact for each elementary operators.

Fix now an operatorL, and F = (f1, . . . , fk) in its definition domain.

By the continuity of L, we find a compact neighborhood U of 0 and a neighborhoodV of 0 such thatLis well defined and continuous on a neigh- borhood W ofF in C^∞(U,R^k).Remark that, if G∈ W is analytic,L(G) is analytic also : each elementary operator preserves analyticity.Suppose now that L = 0.Then, for any analytic G∈ W, L(G) = 0, that means, since L(G) is analytic, thatL(G) = 0.Remind us that analytic functions are dense inC^∞(U).Being continuous and vanishing on a dense subset, L vanishes on allW.Hence, ifL= 0,Lvanishes on a neighborhood of anyF in its definition domain, then everywhere.

3.3. Quasi-analyticity

The two previous lemmas are the main tools to prove that the algebras An(h) are quasi-analytic provided thathis strongly transcendental.

Lemma 3.6. — Leth∈C^∞(U)be strongly transcendental. Then, for all n∈N, the algebra An(h)is quasi-analytic.

Proof. — Fix h ∈ C^∞(U) a strongly transcendental function, n ∈ N, andg∈ An(h) withg= 0.We need to show thatg= 0.

By the definition ofAn(h), there exists an operatorLsuch thatgis the image byLof the germs ofhat some pointsb1, . . . , b ofU:

∃(b1, . . . , b)∈ U\∆, g=L(hb1, . . . , h_b)

where h_bp denote the germ at 0 of t → h(b_p +t).According to lemma 3.4, there exist (a₁, . . . , a_k)∈R^k, a familyP_α of polynomials with integral coefficients, and a family of indexesβ_p,α ∈Nsuch that

∀(f1, . . . , f_j), L(f₁, . . . ,f_j) =

α∈Nⁿ

P_α(a₁, . . . , a_k, j₁^β1,αf₁(0), . . . , j₁^βj,αf_j(0))x^α. Applying this formula to (hb1, . . . , hb), we obtain

0 =g=

α∈Nⁿ

Pα(a1, . . . , ak, j^βαh(b))x^α, (3.2) whereβ_α= max_p=1,...,(β_p,α) andb= (b₁, . . . , b).

Now recall thathis strongly transcendental, and let us make a remark on the algebraic independence of (a1, . . . , ak, j^mh(b)): there exists an integer

N such that the family{h^(p)(b_q); pN+ 1, q= 1, . . . , j} is algebraically independent overQ(a₁, . . . , a_k, j^N h(b)).Indeed, denote byd_mthe transcen- dence degree

d_m= trdeg(a₁, . . . , a_k, j^m h(b)), and recall that there existsC such that,

trdeg(j^mh(b))(m+ 1)−C.

Since dm trdeg(j^mh(b)), we get dm m−C.On the other hand, we have dm−d_m−1.Hence, there are at mostC+d0indexesmsuch that dm−d_m−1< .The researched integerN is the maximum of these indexes.

Let us now express the equality (2) in another way : we set Qα(y1, . . . , y_(βα−N)) =Pα(a1, . . . , ak, j^Nh(b), y1, . . . , y_(βα−N)), where the coefficients of the polynomialsQαbelong toQ[a1, . . . , ak, j^Nh(b)].

The equality (2) becomes:

0 =

α∈Nⁿ

Qα((h^(p)(xq))α)x^α,

where (h^(p)(x_q))_α stands for the (β_α−N)-tuple formed by all h^(p)(b_q) with p = N + 1, . . . , βα and q = 1, . . . , j.Since the families (h^(p)(xq))α

are algebraically independent over the coefficients of the polynomialsQα, it shows that for allα,Qα= 0.In other word,L(f1, . . . ,fj) vanishes as soon asj₁^Nfi(0) =j^N₁ h(bi), fori= 1, . . . , .Hence, if we setN to be the operator

N(f1, . . . , f) =L(Hb^N1 +t^N+1f1, . . . , H_b^N +t^N+1f),

whereH_b^N_i denote the Taylor expansion ofhto the orderN atb_i , we have N = 0.According to lemma 3.5, it follows thatN = 0.Hence we have

g=L(hb1, . . . , hb) =N((hb1−H_b^N₁)/t^N, . . . ,(hb−H_b^N)/t^N) = 0.

Since for any g ∈ An(h), g = 0 imply g = 0, the algebra An(h) is quasi- analytic.

We now prove that Rh is o-minimal if his a restricted transcendantal function.

Proof. —[Proof of theorem 1.2] Let hbe a restricted strongly transcen- dental function.Then there exists ˜ha strongly transcendantal function de- fined in an open neighborhoodU of [0,1], such that ˜h|[0,1] =h.The algebras

An(˜h) are quasi-analytic by lemma 3.6, so, according to theorem 3.3,RHis o-minimal, polynomially bounded and admits C^∞-cell decomposition, where His the collection of all restrictions of ˜hto closed intervals included inU. SinceRh is a reduct ofRH, theorem 1.2 holds.

4. Corollaries

In this section, we apply the above theorems to give new proofs of some results on o-minimality.In [LGR08], it is asked if the method developed there allows to construct an o-minimal structure that does admit smooth cell decomposition, but not analytic cell decomposition, the main interest of such a proof would be to circumvent the use of the Mandelbrojt’s theorem needed in [RSW03].In one sense, we answer here by the positive.Actually, we do not construct such a structure, but prove its existence.

Corollary 4.1. — There exists a residual subset T of restricted C^∞- function such that, for all functionhinT,Rhis o-minimal, admits C^∞-cell decomposition, but does not admit analytic cell decomposition.

Proof. — It suffices to recall that the set of restricted C^∞-functions that are nowhere analytic is residual.Hence, its intersection T with re- stricted strongly transcendental functions is residual, and has the required propreties.

We also obtain the existence of non compatible o-minimal structures.

The interest of this new proof is again to circumvent the use of the Man- delbrojt’s theorem needed in [RSW03].

Corollary 4.2. — There exist h1 and h2 such that Rh1 and Rh2 are o-minimal, but Rh1,h2 is not o-minimal.

Proof. — Fixha restricted C^∞-function.We claim that there exist two restricted strongly transcendental functions h₁, h₂ such that h=h₁+h₂. Indeed, letψdenotes the map, defined on C^∞([0,1]) byψ(g) =h−g.This map conserves genericity.Hence, the intersectionψ(st)∩stis residual, then non empty.A functionh₁belongs to this intersection if there existsh₂inst such thatψ(h₂) =h₁, this meansh=h₁+h₂.The corollary then stands by choosing anhsuch thatRhis not o-minimal :Rh1 andRh2 are o-minimal, since h1 and h2 are strongly transcendantal, while Rh1,h2 is not, since it definesh.

One can improve the last, askingh2to be analytic.It gives the following.

Corollary 4.3. — There exists a functionhsuch thatRhis o-minimal, whileRh,anis not o-minimal.

Proof. — Choose an R > 0, and a C^∞-functionf on a neighborhood U of [0,1], such that Rf|[0,1] is not o-minimal.According to proposition 2.2, the set of strongly transcendental functions is residual in f +AR, in particular not empty.So, there exists a strongly transcendantal function h∈C^∞(U) such that f−his analytic.Defineh=h|[0,1].The structure Rh is o-minimal, since his strongly transcendental, butRh,an is not, since it definesf|[0,1].

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